1. Delaunay Triangulation on the GPU
Dan Maljovec

2. CPU Delaunay Triangulation
Randomized Incremental Algorithm
Construct Bounding triangle
Choose point to insert randomly
Locate triangle containing point
Construct new children triangles by connecting new point to each point of containing triangle
Test edges and flip if necessary
If flipped, test edges and recursively flip if necessary
Repeat 2-5 until all points have been inserted

3. GPU Implementation Use the dual: Voronoi Diagram
Construct a discrete Voronoi Diagram (Jump Flooding Algorithm)
Obtain a triangulation from the Voronoi
Fix the estimations we made and perform edge flipping

4. Jump Flooding Algorithm
Write the sites into a square texture
Map each pixel in the texture (a point in discrete space) to a thread
Initialize the step size to be half of the width/height of the texture
Each pixel will look in 8 directions of size step to find closest site to it
Divide the step size by 2 and Repeat

5. 1+JFA Write the sites into a square texture
Map each pixel in the texture (a point in discrete space) to a thread
Each pixel will look in eight directions of size step to find closest site to it
Divide the step size by 2 and Repeat
First do one round of step = 1,
then step = textureDimension / 2

6. 1+JFA (write sites)

7. 1+JFA (step size = 1)

8. 1+JFA (step size = 1)

9. 1+JFA (step size = 8)

10. 1+JFA (step size = 8)

11. 1+JFA (step size = 4)

12. 1+JFA (step size = 4)

13. 1+JFA (step size = 2)

14. 1+JFA (step size = 2)

15. 1+JFA (step size = 1)

16. Errors from JFA
JFA can introduce errors in the Voronoi diagram, which would lead to crossing triangles. These errors are manifested in “island” pixels as shown below:

17. Detect and Remove Islands
Map pixels to threads
Knowing where a site is with respect to a pixel, we can look in that direction and see if this pixel is stranded
Iterate this process until no more islands are found

18. Detect and Remove Islands
Map pixels to threads
Knowing where a site is with respect to a pixel, we can look in that direction and see if this pixel is stranded
Iterate this process until no more islands are found
This pixel’s site is up and to the right, so we will check neighbors in those directions to see if this is an island.

19. Detect and Remove Islands
Map pixels to threads
Knowing where a site is with respect to a pixel, we can look in that direction and see if this pixel is stranded
Iterate this process until no more islands are found
Update it to the closest site of the colors we find at the neighboring pixels

20. Find Voronoi Vertices
Vertices in the discrete Voronoi diagram, lie at junctions of 3 or 4 different colored pixels. Map pixels to threads, and look right, down, and diagonally down and to the right to test if there exists a vertex at this pixel.
3-Color Cases
(generates 1 Δ):
4-Color Case
(generates 2 Δ):
Pixels will be marked as Voronoi vertices

21. Chain Up Voronoi Vertices
Parallel Prefix operation:
Map rows to threads, and for each pixel store the closest vertex in the row, to its right.
Allows us to apriori know how many triangles we are about to create, and allows us to be output sensitive in the next step.

22. Chain up Vertices 0 2 1 2 #Vertices Row Δ to Generate Row -1 -1 -1 -12 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 3 3 3 5 5 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 10 10 10 10 10 10 10 10 10 10 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1

23. Triangle Generation Map thread to each row
Create a triangle at each 3-Color vertex by connecting the 3 sites that contribute to the vertex
Create 2 triangles at each 4-Color vertex by connecting sites of pixels 1, 2, & 3, then sites of pixels 2,4, & 3 1 3 2 4

24. Generate Triangles

25. Generate Triangles

26. Generate Triangles

27. Generate Triangles

28. Generate Triangles

29. Generate Triangles

30. Generate Triangles

31. Fix Convex Hull
What’s wrong here? Vertex will not be captured in texture, but it does exist.
The bisectors will cross eventually if the points are not convex. Sites that form a convex turn will form bisectors that extend away from each other as they go to infinity.

32. Fix Convex Hull
What’s wrong here? Vertex will not be captured in texture, but it does exist. Is there a specific case where this occurs? Moving clockwise around cells who touch the border, sites that form a left turn will have a vertex outside the texture
The bisectors will cross eventually if the points are not convex. Sites that form a convex turn will form bisectors that extend away from each other as they go to infinity.

33. Shift Sites
We shifted all the sites into discrete space, now we have to put them back without causing any intersecting triangles

34. Shift Sites

35. Shift Sites
1st case, we relocate to the inside of a triangle that lies within the fan without crossing any edges

36. Shift Sites
2nd case, we relocate to the inside of a triangle that lies within the fan but we cause edges to cross

37. Shift Sites
3rd case, we relocate to the inside of a triangle outside of the triangle fan connected to the site

38. Shift Sites
4th case, we relocate outside of the mesh

39. Shift Sites

40. Insert Missing Sites
Similar to what we did with the incremental approach, point locate a site in a triangle and subdivide it, but since we know who took the missing site’s position, we can start with the triangle fan of that point.

41. Insert Missing Sites
1st case, point lies in the interior of a triangle

42. Insert Missing Sites
2nd case, point lies on an edge

43. Insert Missing Sites
3rd case, point lies outside of the mesh

44. Insert Missing Sites

45. Flip Edges
We have seen this before
Empty Circumcircle test

46. Flip Edges
We have seen this before
Empty Circumcircle test

47. Flip Edges
We have seen this before
Empty Circumcircle test

48. Flip Edges
We have seen this before
Empty Circumcircle test

49. Flip Edges
We have seen this before
Empty Circumcircle test

50. Summary 1+JFA Island removal Find vertices Generate Δs Fix convex hull
Shift sites
Insert missing sites
Flip edges

51. References
Hoff, K. E., Keyser, J., Lin, M., Manocha, D., Culver, T Fast Computation of Generalized
Voronoi Diagrams Using Graphics Hardware. In Proceedings of ACM SIGGRAPH 1999,
ACM Press / ACM SIGGRAPH, New York. Computer Graphics Proceedings, Annual
Conference Series, ACM,
Denny, M. O Algorithmic geometry via graphics hardware. PhD thesis, Universität des
Saarlandes.
Rong, G., Tan, T.-S Jump flooding in GPU with applications to Voronoi diagram and
distance transform. In Proceedings of the Symposium on Interactive 3D Graphics and
Games, ACM Press, 109–116.
Rong, G., Tan, T.-S Variants of jump flooding algorithm for computing discrete Voronoi
diagrams. In Proceedings of the 4th International Symposium on Voronoi Diagrams in
Science and Engineering (ISVD’07), 176–181.
Rong, G., Tan, T.-S., and Cao, T.-T. and Stephanus Computing Two dimensional Delaunay
Triangulation Using Graphics Hardware. In Proceedings of the Symposium on
Interactive 3D Graphics and Games, ACM Press, 89–97.

59. Chain up Vertices
#Vertices
Row
Δ to Generate
Row -1 8 6
0 1 1